Entanglement, correlations, and the energy gap in many-body quantum systems

What correlations are present in the ground state of a many-body Hamiltonian? We study the relationship between ground-state correlations, especially entanglement, and the energy gap between the ground and first excited states. We prove several general inequalities which show quantitatively that ground-state correlations between systems not directly coupled by the Hamiltonian necessarily imply a small energy gap.